A sub-division-ring is a subset of a division ring's set which is a
division ring under the induced operation. If the overring is
commutative this is a field; no special consideration is made of the
fields in the center of a skew field. (Contributed by Stefan O'Rear,
3-Oct-2015.)

Example of the Fundamental Theorem of Calculus, part two (ftc219386):
. Section 4.4
example 1a of [LarsonHostetlerEdwards] p. 311.
(The book teaches ftc219386
as simply the "Fundamental Theorem of Calculus", then ftc119384
as the
"Second Fundamental Theorem of Calculus".) (Contributed by
Steve
Rodriguez, 28-Oct-2015.) (Revised by Steve Rodriguez, 31-Oct-2015.)

The derivative of a function on is zero iff it is a constant
function. Roughly a biconditional analog of dvconst19261 and
dveq019342. Corresponds to integration formula
" " in
section 4.1 of [LarsonHostetlerEdwards] p. 278.
(Contributed by Steve
Rodriguez, 11-Nov-2015.)

Exponential growth and decay model. The derivative of a function y of
variable t equals a constant k times y itself, iff
y equals some
constant C times the exponential of kt. This theorem and
expgrowthi26961 illustrate one of the simplest and most
crucial classes of
differential equations, equations that relate functions to their
derivatives.

Section 6.3 of [Strang] p. 242 calls
y' = ky "the most important
differential equation in applied mathematics". In the field of
population ecology it is known as the Malthusian growth model or
exponential law, and C, k, and t correspond
to initial
population size, growth rate, and time respectively
(https://en.wikipedia.org/wiki/Malthusian_growth_model);
and in
finance, the model appears in a similar role in continuous
compounding
with C as the initial amount of money. In exponential
decay models,
k is often expressed as the negative of a positive constant
λ.

Here y' is given as , C as , and ky as
.
is the constant
function that maps any real or complex input to k and is
multiplication as a function operation.